Commutativity in a Poincaré-Type Inequality
Abstract
We study a Poincaré-type inequality for Gaussian measures restricted to origin-symmetric convex sets. The inequality arises from the second variation at t=0 of the function tlongmapsto γΣ(etAK), where Σ and A are symmetric matrices. The relevance of this function comes from Saroglou's reformulation of the log-Brunn--Minkowski conjecture. We show that Σ and A commute if and only if the inequality holds for every box of the form UD B∞n, where U is orthogonal and D is diagonal. In the non-commuting case, this yields counterexamples for orthogonal boxes whose axes can be chosen arbitrarily close to the coordinate axes. We also show that the inequality holds when K is unconditional with respect to an eigenbasis of Σ and A is an arbitrary symmetric matrix.
Related
- A fractional modular Poincaré--Wirtinger-type inequality in Sobolev spaces with variable exponents
- A priori bounds for geodesic diameter. Part III. A Sobolev–Poincaré inequality and applications to a variety of geometric variational problems
- Sharp Poincaré inequality for W02,p
- Any uniformly rectifiable set is contained in a set with Poincaré inequality
- Poincaré inequality and quantitative De Giorgi method for hypoelliptic operators
BibTeX
@misc{kalantzopoulos2026commutativity,
title = {{Commutativity in a Poincaré-Type Inequality}},
author = {Pavlos Kalantzopoulos},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07248},
eprint = {2610.07248},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07248},
}
RIS
TY - UNPB
TI - Commutativity in a Poincaré-Type Inequality
AU - Pavlos Kalantzopoulos
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.07248
UR - https://arxiv.org/abs/2610.07248
AB - We study a Poincaré-type inequality for Gaussian measures restricted to origin-symmetric convex sets. The inequality arises from the second variation at t=0 of the function tlongmapsto γ_Σ(e^tAK), where Σ and A are symmetric matrices. The relevance of this function comes from Saroglou's reformulation of the log-Brunn--Minkowski conjecture. We show that Σ and A commute if and only if the inequality holds for every box of the form UD B_∞^n, where U is orthogonal and D is diagonal. In the non-commuting case, this yields counterexamples for orthogonal boxes whose axes can be chosen arbitrarily close to the coordinate axes. We also show that the inequality holds when K is unconditional with respect to an eigenbasis of Σ and A is an arbitrary symmetric matrix.
ER -